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Name _________________
Common Core Geometry R
A
1)
Date ___________________
HW #21 Triangle Proofs
B
Given: AD ≅ BC
E is the midpoint of DC
AD
DC
BC
DC
Prove: ΔADE ≅ ΔBCE
D
E
Statements
1. AD ≅ BC, E is midpoint of DC.
2. DE ≅ CE
3. AD DE, BC DC
4. ≮D and ≮C are right angles.
5. ≮D ≅ ≮C
6. ΔADE ≅ΔBCD
C
Reasons
1. Given
2. A midpoint divides a segment into 2 ≅
parts.
3. Given
4. Perpendicular lines form right angles.
5. Right angles are ≅.
6. SAS
B
2)
A
C
Given: AC bisects ≮BCD
BC ≅ CD
Prove: ΔABC ≅ ΔADC
D
Statements
1. AC bisects ≮BCD.
2. ≮BCA ≅ ≮DCA
3. BC ≅ CD
4. AC ≅ AC
5. ΔABC ≅ ΔADC
Reasons
1. Given
2. An angle bisector divides an angle into
two congruent parts.
3. Given
4. Reflexive Property
5. SAS
OVER
3) D
A
E
C
B
Statements
1. CD bisects AB.
2. AE ≅ BE
Prove: ΔAEC ≅ ΔBED
Reasons
1. Given
2. A segment bisector divides the segment
into two ≅ parts.
3. Given
4. Vertical angles are ≅.
5. ASA
3. ≮CAE ≅ ≮DBE
4. ≮AEC ≅ ≮BED
5. ΔAEC ≅ ΔBED
4) Given: CD bisects AB
≮CAE ≅ ≮DBE
B
Given: BD is the perpendicular bisector of AC
Prove: ΔABD ≅ ΔCBD
A
D
C
Statements
Reasons
1. BD is the perpendicular bisector of 1. Given
AC.
2. ≮ADB and ≮CDB are right angles. 2. Perpendicular lines form right angles.
3. ≮ADB ≅ ≮CDB
3. Right angles are ≅.
4. AD ≅ CD
4. A bisector divides an angle into 2 ≅
parts.
5. BD ≅ BD
5. Reflexive Property
6. ΔABD ≅ ΔCBD
6. SAS
Aim 22: Proving Triangles Congruent by HL, and using Supplementary Angles
Do Now:
a) If m≮EBC = 110 and m≮BCF = 110, find m≮ABE and m≮DCF.
F
E
b) What do you notice about your answers in (a)?
B
A
C
c) What can we conclude about supplements of equal angles?
Hypotenuse- Leg Triangle Congruence criteria (HL) : Given two right triangles
ABC and A'B'C' with right angles B and B'. If AB = A'B' (leg) and AC = A'C'
(hypotenuse), then the triangles are congruent.
B
B
A
C
A
A'
C
C'
B'
B'
1.
Given: PQ
QR, PS
Prove: ΔPQR ≅ ΔPSR
Statements
Q
SR, QR ≅ SR
R
P
Reasons
S
D
2.
Given: SM
RT, ≮PRM ≅ ≮QTM
Prove: ΔSRM ≅ ΔSTM
Statements
Reasons
3. Given: AB ≅ CD, AB ll CD
Prove: ΔABC ≅ ΔDAC Statements
Reasons
4. A
E
Given: ≮ABE ≅ ≮FCD
BF || EC
C
B
D
Prove: ΔBEC ≅ ΔCFB
F
Statements
Reasons
C
5. Given: AB and CD intersect at E. AC ll BD.
E is the midpoint of CD.
Prove: ΔACE ≅ Δ BDE
B
E
A
D
Statements
Reasons
6.
B
A
D
C
E
Prove:
ΔADE ≅ ΔBCE
Reasons
Statements
7.
Given: AE ≅ BE
E is the midpoint of DC
AD
DC
BC
DC
C
Given: ≮CBE ≅ ≮DBE
A
B
E
AE bisects ≮CAD
Prove: ΔACB ≅ ΔADB
D
Statements
Reasons
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